Markov Chain Monte Carlo is a widely used tool in applied mathematics, with rates of convergence being a key aspect: how long must a chain be run to fulfil its purpose? This includes questions such as "How many times should a deck of cards be shuffled to achieve a proper mix?" There has been a close interplay between these problems and classical analysis, with Poincaré, Nash, Sobolev, Harnack, and log-Sobolev inequalities playing a crucial role. Moreover, finite analogues of concepts such as Whitney covers, John domains, and inner uniform domains allow for sharp answers to real-world questions.